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\title{高等代数一}
\subtitle{16-习题与问答-分块矩阵-向量空间 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
%\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
\date{{\ppr 2022年11月15日} }

\maketitle

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\begin{enumerate}

\item  相抵标准形的应用
\item  行列式乘积公式的应用
\item  分块行初等变换
\item  复数的运算
\item  向量空间的概念
\end{enumerate}


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\begin{frame}{讲解本次作业的同学 }

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{\small 
\begin{table}[ht]
\centering
\begin{tabular}{cccccc}
4-习题&8-习题&12-习题&16-习题&20-习题&24-习题 \\ \hline 
{01}&{02}&03&\underline{04}&05&06 \\   
{07}&{08}&09&\underline{10}&11&12 \\  
{13}&{14}&15&\underline{16}&17&18 \\ 
{19}&{20}&21&\underline{22}&23&24 \\  
{25}&{26}&27&\underline{28}&29&30 \\  
{31}&{32}&33&\underline{34}&35&36 \\  
{37}&{38}&39&\underline{40}&41&42 \\  
{43}&{44}&45&\underline{46}&47&48 \\ 
{49}&{50}&51&\underline{52}&53&54 \\  
\end{tabular}
\end{table}
}

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\begin{itemize}

\item  习题1：证明一个秩为 $r$ 的矩阵可以写成 $r$ 个秩为1的矩阵的和。

\item  解答思路：考虑相抵标准形。


\end{itemize}

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\begin{itemize}

\item  习题2：设有二阶实数矩阵 $A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}$, 且存在正整数 $m$ 使得 $A^m=O$ 为零矩阵。证明 $A^2=O$. 

\item  解答思路：一种方法是使用行列式乘积公式，证明矩阵 $A$ 的秩是0或1.  


\end{itemize}

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\begin{itemize}

\item  习题3：设 $A$ 是幂零矩阵，即存在正整数 $m$ 使得 $A^m=O$ 为零矩阵。求矩阵 $X$ 使得 $AX+A+X=O$.  

\item  解答思路：考虑 $E+A$ 的逆阵。


\end{itemize}

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\begin{itemize}

\item  习题4：设 $A,B,C,D$ 是 $n$ 阶矩阵。设 $\det(A)\neq 0$ 且 $AC=CA$. 设 $2n$ 阶矩阵 $M=\begin{pmatrix} A&B \\ C&D \end{pmatrix}$. 证明 $\det(M)=\det(AD-CB)$. 

\item  解答思路：使用分块初等变换。


\end{itemize}

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\begin{itemize}

\item  习题5：设 $V$ 是实向量空间。设 $\alpha_1,\alpha_2\in V$ 并记向量组 $\Phi=(\alpha_1,\alpha_2)$. 
设 $A=\begin{pmatrix} k&\ell \\ m&n \end{pmatrix}$ 是一个实数矩阵。定义如下的线性运算
\begin{eqnarray*}
\Phi \cdot A=
\begin{pmatrix} \alpha_1 & \alpha_2 \end{pmatrix}
\begin{pmatrix} k&\ell \\ m&n \end{pmatrix} 
=
\begin{pmatrix} k\alpha_1+m\alpha_2 & \ell\alpha_1+n\alpha_2 \end{pmatrix}.
\end{eqnarray*}
设 $B$ 是另一个二阶实数矩阵。证明 $(\Phi \cdot A)\cdot B=\Phi \cdot (AB)$. 

\item  解答思路：将矩阵 $B$ 的元素也写出来，然后分别计算等式两边。


\end{itemize}

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\begin{itemize}

\item  习题6：设 $V=\mathbb{R}[x]_2$ 是次数不超过2的实系数多项式全体组成的集合，即 $V$ 中的一般元素可以写成 $\alpha=a_0+a_1x+a_2x^2$, 其中 $a_0,a_1,a_2$ 是实数。 设 $V$ 中的另一个元素为 $\beta=b_0+b_1x+b_2x^2$, 设 $k\in\mathbb{R}$. 定义 $V$ 中元素的加法与数乘分别为
\begin{eqnarray*}
\alpha+\beta &=& (a_0+b_0) + (a_1+b_1)x + (a_2+b_2)x^2, \\
k\cdot\alpha&=& ka_0+ka_1x+ka_2x^2. 
\end{eqnarray*}

\vspace{-0.2cm}

\begin{enumerate}
\item  证明 $V$ 在这两个运算下是一个实向量空间。
\item  设 $\eta_0=1, \eta_1=x-1, \eta_2=(x-1)^2$. 将 $\alpha=3+4x+5x^2$ 写成 $k_0\eta_0+k_1\eta_1+k_2\eta_2$ 的形式，其中 $k_0,k_1,k_2$ 是待定的实数。
\end{enumerate} 

\item  解答思路：根据两个多项式相等的定义是每个 $x^i$ 前面的系数对应相等，可以列出关于 $k_0, k_1, k_2$ 的三个方程。


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\begin{itemize}

\item  习题7：练习复数的运算：
\begin{enumerate}
\item  $(4+i)(5+3i) - (3+i)(3-i)$. 
\item  $(1+i)^{100}$.
\item  解方程 $z^2-5z+4+10i=0$.
\item  解方程 $z^{10}=512(1-\sqrt{3}i)$. 
\item  求当 $|z|\le 1$ 时 $|3+2i-z|$ 的最小值。
\end{enumerate} 

\item  解答思路：使用直角坐标或极坐标来表示复数。


\end{itemize}

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\begin{itemize}

\item  习题8：设 $V$ 为正实数全体组成的集合。设 $x,y\in V$, 设 $k\in\mathbb{R}$. 定义 $V$ 中的元素的“加法”与“数乘”分别为
\begin{eqnarray*}
%\alpha \oplus \beta &=& \alpha\beta, \\
%k\odot\alpha&=& \alpha^k. 
x \oplus y &=& xy, \\
k\odot x&=& x^k, 
\end{eqnarray*}
其中等号右边是通常的实数乘积与幂次。
\begin{enumerate}
\item  计算 $2\odot (3\oplus 4)$. 
\item  证明 $(V,\oplus,\odot)$ 是实数域上的一个向量空间。 
\end{enumerate} 

\item  解答思路：首先验证这两个运算的结果仍然在集合 $V$ 中。然后验证八条公理。


\end{itemize}

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\begin{itemize}

\item  习题9：设 $V$ 是所有极限为零的实数数列组成的集合，即 $V$ 中的一般元素为 $\alpha=(x_1,x_2,\cdots, x_n,\cdots)$, 其中 $x_i$ 都是实数，且 $\lim\limits_{n\to\infty} x_n=0$. 
设 $\beta=(y_1,y_2,\cdots, y_n,\cdots)\in V$, 设 $k\in\mathbb{R}$. 定义 $V$ 中的加法和数乘分别为
\begin{eqnarray*}
\alpha+\beta &=& (x_1+y_1, x_2+y_2, \cdots, x_n+y_n,\cdots), \\
k\cdot\alpha&=& (kx_1,kx_2,\cdots, kx_n,\cdots). 
\end{eqnarray*}
证明 $V$ 在这两个运算下是一个实向量空间。

\item  解答思路：首先验证 $\alpha+\beta$ 与 $k\cdot\alpha$ 仍是集合 $V$ 中的元素。这需要用到极限的性质。


\end{itemize}

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\begin{itemize}

\item  习题10：设 $A,B,C$ 是 $n$ 阶矩阵，设 $A,C$ 是可逆矩阵。记 $O$ 是 $n$ 阶零矩阵。设 $2n$ 阶矩阵 $M=\begin{pmatrix} A&O \\ B&C \end{pmatrix}$. 计算 $\det(M)$ 与伴随矩阵 $M^*$. 

\item  解答思路：


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